Integrate dx/2-x^2 =?

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Answer 1 · 2018-04-26T18:25:15

#=1/sqrt2ln(sqrt2/sqrt(2-x^2)+x/sqrt(2-x^2))+C#

#intdx/(2-x^2)#

Using Trigonometric substitution

#x=sqrt2sinu#

#dx=sqrt2cosu*du#

Substitute

#int(sqrt2cosu*du)/(2-2sin^2u)#

#color(green)(2-2sin^2u=2cos^2u)#

#=int(sqrt2cosudu)/(2cos^2u#

Simplify

#=1/sqrt2int(du)/cosu#

#color(green)(1/cosu=secu)#

#=1/sqrt2intsecudu#

#=1/sqrt2ln(secu+tanu)+C#

Reverse the Substitution

#=1/sqrt2ln(sqrt2/sqrt(2-x^2)+x/sqrt(2-x^2))+C#